Question

Estimate the instantaneous rate of change of P ( x ) = 3 x 2 + 5 at the point x = − 2 In other words, choose x-values that are getting closer and closer to − 2 and compute the slope of the secant lines at each value. Then, use the trend/pattern you see to estimate the slope of the tangent line.

Answer 1

Answer:

The slope of the tangent line is -12

Step-by-step explanation:

Instantaneous Rate of Change

We can approximate to the value of the slope of one function P(x) by computing the slope of the secant line as follows

$\displaystyle m=\frac{P(x+\Delta x)-P(x)}{\Delta x}$

Where $\Delta x$ is an infinitesimal change of x, as small as we want. Our function is

$\displaystyle P(x)=3x^2+5$

We are required to find the instantaneous rate of change in x=-2 by iteratively getting close to it and estimating the slope according to the observed trend.

Let´s use

$\displaystyle \Delta x=0.1$

Then the approximate slope of P in x=-2 is

$\displaystyle m=\frac{P(-2+0.1)-P(-2)}{0.1}$

$\displaystyle m=\frac{P(-1.9)-P(-2)}{0.1}$

We compute  

$\displaystyle P(-1.9)=3(-1.9)^2+5=15.84$

$\displaystyle P(-2)=3(-2)^2+5=17$

Replacing in the slope

$\displaystyle m=\frac{15.83-17}{0.1}=-11.7$

Now we use a smaller infinitesimal or differential

$\displaystyle \Delta x=0.01$

$\displaystyle m=\frac{P(-2+0.01)-P(-2)}{0.01}$

$\displaystyle m=\frac{P(-1.99)-P(-2)}{0.01}$

$\displaystyle P(-1.99)=3(-1.99)^2+5=16.88$

$\displaystyle m=\frac{16.88-17}{0.01}=-11.97$

We can see the slope is getting closer to -12 as the infinitesimal tends to 0, thus we can estimate the slope of the tangent line is -12